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Chapter 19: Problem 10

In the \(x\)-ray tube before striking the target we accelerate the electrons through a potential difference of \(V\) volt. For which of the following value of \(V\), we will have \(x\)-rays of largest wavelength? (A) \(10 \mathrm{kV}\) (B) \(20 \mathrm{kV}\) (C) \(30 \mathrm{kV}\) (D) \(40 \mathrm{kV}\)

Short Answer

Expert verified
The correct option for the potential difference that results in the largest X-ray wavelength is (A) $10 \mathrm{kV}$.

Step by step solution

01

Calculate the energy of the electrons after acceleration

The energy of the accelerated electrons can be found using the formula: \(E = eV\) where E is the energy of the electrons, e is the charge of an electron (\(1.6 \times 10^{-19} C\)), and V is the potential difference in volts.
02

Determine the energy-to-wavelength conversion formula

We can convert the electron energy (E) to X-ray wavelength (\(位\)) using the Planck's equation: \(E = h\frac{c}{\lambda}\) where \(h = 6.626 \times 10^{-34} Js\) is the Planck's constant and \(c = 3 \times 10^8 m/s\) is the speed of light in a vacuum.
03

Calculate the wavelength for each potential difference

Rearrange the Planck's equation to solve for wavelength: \( \lambda = \frac{hc}{E} \) For each potential difference, we will substitute the energy E calculated in step 1 and find the corresponding wavelength: (A) For \(V = 10kV\), \(E = eV = (1.6 \times 10^{-19}C)(10^4V) \) \( \lambda = \frac{hc}{eV} = \frac{(6.626 \times 10^{-34}Js)(3 \times 10^8m/s)}{(1.6 \times 10^{-19}C)(10^4V)} \) (B) Repeat this calculation for \(V = 20kV\), \(V = 30kV\), and \(V = 40kV\).
04

Find the largest wavelength among the calculated values

After calculating the wavelength for each potential difference, we need to compare the values and find the largest wavelength value. The largest wavelength corresponds to the potential difference that will produce X-rays with the largest wavelength. Answer: Based on the calculations and comparison, select the correct option (A, B, C or D) as the potential difference that results in the largest X-ray wavelength.

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Most popular questions from this chapter

Carbon, silicon and germanium have four valence electrons each. At room temperature which one of the following statements is most appropriate? (A) The number of free electrons for conduction is significant only in Si and Ge but small in \(C\) (B) The number of free conduction electrons is significant in \(C\) but small in Si and Ge (C) The number of free conduction electrons is negligibly small in all the three (D) The number of free electrons for conduction is significant in all the three.

If the binding energy of the electron in a hydrogen atom is \(13.6 \mathrm{eV}\), the energy required to remove the electron from the first excited state of \(\mathrm{Li}^{++}\)is (A) \(30.6 \mathrm{eV}\) (B) \(13.6 \mathrm{eV}\) (C) \(3.4 \mathrm{eV}\) (D) \(122.4 \mathrm{eV}\)

In the nuclear nuclei $$ { }_{1}^{2} \mathrm{H}+{ }_{1}^{3} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}+n $$ given that the repulsive potential energy between the two nuclei is \(-7.7 \times 10^{-14} \mathrm{~J}\), the temperature at which the gases must be heated to initiate the reaction is nearly [Boltzmann's constant \(\left.k=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}\right]\) [2003] (A) \(10^{7} \mathrm{~K}\) (B) \(10^{5} \mathrm{~K}\) (C) \(10^{3} \mathrm{~K}\) (D) \(10^{9} \mathrm{~K}\)

Which of the following statement about \(x\)-rays is/are true? (A) \(E\left(K_{\alpha}\right)+E\left(L_{\beta}\right)=E\left(K_{\beta}\right)+E\left(M_{\alpha}\right)=E\left(K_{\gamma}\right)\), where \(E\) is the energy of respective \(x\)-rays. (B) For the harder \(x\)-rays, the intensity is higher than soft \(x\)-rays. (C) The continuous and the characteristic \(x\)-rays differ only in the method of creation. (D) The cut-off wavelength \(\lambda_{\min }\) depends only on the accelerating voltage applied between the target and the filament.

A proton of mass \(m\) and charge \(+\mathrm{e}\) is moving in a circular orbit in a magnetic field with energy \(1 \mathrm{MeV}\). What should be the energy of \(\alpha\)-particle (mass \(=4 \mathrm{~m}\) and charge \(=+2 \mathrm{e}\) ), so that it can revolve in the path of same radius (A) \(1 \mathrm{MeV}\) (B) \(4 \mathrm{MeV}\) (C) \(2 \mathrm{MeV}\) (D) \(0.5 \mathrm{MeV}\)
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